U. S. Department of Commerce National Bureau of Standards Research Paper RP2059 Volume 44, January 1950 Part of the Journal of Research of the National Bureau of Standards Solution of the Telegraphers Equation With Boundary Conditions on Only One Characteristic ] By George E. Forsythe Forecasting a certain idealized horizontal, autobarotropic, nonviscous, nondiverging atmospheric flow considered by Rossby leads to an unusual boundary-value problem for the telegrapher's equation, involving boundary values on only one characteristic. It is shown how to find unique solutions periodic in the longitude; these are represented in terms of a Green's function, A procedure for computing the Green's function is set down and is shown to be optimal in a restricted sense. The Green's function is tabulated for 72 longi- tudes and 14 time-values. An alternative solution by a difference equation is mentioned. I. Introduction In one treatment of planetary atmospheric flow as horizontal, autobarotropic, nonviscous, and nondiverging in a plane, Rossby [9] 2 considered the idealized case of a constant west-wind com- ponent, U, and a south-wind component, v, depend- ent on the west-to-east distance coordinate,^, and time, r, but independent of the south-to-north distance coordinate, /x. It was shown in [5] that this v satisfies the telegrapher's equation (eq 3) below, where x=\[/—Ut, f=4 fir. The parameter £= 2Q cos <p (d<p /did) is here considered constant; 12 is the angular speed of the earth's rotation, and <p is latitude. For this simple atmospheric model, the meteorological forecast problem is one of deter- mining v(x, t) for future times t, given only v(x f 0). But on a plane the specification of v(x, 0) is not sufficient to determine v(x, t) for many £>0, be- cause the line t—0 is a characteristic of eq 3 (see p. 254 of [11]). Having its initial conditions on only one characteristic is an unusual feature of the present problem that does not seem to have arisen in other physical problems known to the author to lead to the telegrapher's equation. The author shows that the forecast problem has a unique solution when it is assumed that the world is round, that is, when the solution is assumed to 1 This paper was written at the Institute for Numerical Analysis of the National Bureau of Standards with the financial support of the Oflice of Naval Research of the U. S. Navy Department. 2 Figures in brackets indicate the literature references at the end of this paper. be periodic in x. The problem is stated in section II and solved in section III. In section IV the solution is represented in terms of a Green's func- tion. In section V a procedure is outlined for computing the Green's function by improving the convergence of its Fourier series. In section VI certain auxiliary polynomials, a k (x), used in seel ion V are discussed and related to the Bernoulli polynomials. In section VII are reported without proof a few results on the approximate solution of the problem by a difference equation, taken from [(>]. In section VIII is given a table of values of the Green's function, as computed in the Compu- tation Unit of the Institute for Numerical Analysis. The present author first reported this work in [7]. Independently of the research reported here, Charney, Eliassen, and Hunt of the Institute for Advanced Study considered the telegrapher's equation while investigating numerical weather prediction in general. Their research was reported in [1] and is written up in [2]. The work of these men includes much of what is reported here, and much more. II. Statement of the Problem Let C be the circumference of a unit circle; let us adopt an angle coordinate x for C\ — tt<^x<tt. Let / be the set of time-instants t: 0<t<^ °° . Let R be the closed two-dimensional region consisting of all points (x, t) with x in C and t in /. Let f(x) be a real-valued function that satisfies the follow- Solution of the Telegrapher's Equation 89 ing hypotheses, but which is otherwise arbitrary: Hi. f(x) is sectionally smooth 3 on C. Moreover, f(x)=i[f(x+0)+f(x-0)], (all x). H 2 : fix) has the average value zero: f(r)dx=0. /: (i) (2) The problem is to find a real-valued function, v(x, t), defined everywhere on i?,with the following four properties: Pi'. v t exists* and is continuous throughout R. P 2 : v x and v tx =c)Vt/c>x exist and are continuous everywhere in R except, at most, for a finite number of values of 5 x. Pz'. Whenever v xt is defined, the following hyper- bolic partial differential equation (the telegrapher's equation) is satisfied: v xt +lv=0. (3) P 4 : For t=0, v(x, t) reduces tof(x): »(3,0)=/(3). III. Solution of the Problem, Uniqueness One gets a formal solution by separation of variables 'and use of Fourier series. Assume a solution of eq 3 of form v(x, t) =X(x) T{t) . Then v x .=X'(x)T'(t), and eq 3 takes the form X'V) T'{t) X(x) ' T(t) - (4) The two factors in eq 4 must themselves be constant: X'(x) , X(x) T'(t) _ T(t) _1_ "4\* (5) (6) A solution of eq 5 for — oo<^x<oo is X(x)=e Kx . For x on the circle (7, however, one must have X(—Tr)=X(ir), or e"* x =€* x . Taking logarithms, oneseesthat — 7rX=7rX+2n^(?i=0, ±1, ±2, . . .). Hence X=m(7i=0, ±1, ±2, . . .). Since the value 3 That is, both f(x) and/'(z) are continuous in C except for a finite number of jump discontinuities. 4 The subscripts denote partial derivatives. » It is shown on pp. 55 to 57 of [3] that our conditions Pi and P 2 imply the following: r xt exists and equals Vt x for all (x, t) such that x is not one of the excepted values in P 2 . X=0 is incompatible with eq 4, there remain the following fundamental solutions of eq 5: X n (x)-- (n=±l,±2, ±3, .)■ Corresponding to X n (x), the solution T n (t) of eq 6 for \=ni is T n (t) = exp (it/4n) . Hence for n= ± 1 , ±2, . . . the functions X n (x)T n (f)=exp [i(nx+ t/4:n)] have properties P u P 2 , and P 3 . By taking linear combinations of the functions X n T n and X_ n T_ nj one obtains the equivalent pair of func- tions cos (nx-\-t/4:n) and sin (nx-\-t/4ri). Both of the latter functions have properties P l9 P 2 , and P 3 . In order to obtain a solution with enough de- grees of freedom to satisfy P 4 , consider the series, v(x,f)' -§ [ a » cos ( nx+ L) +b " sin ( w+ ^)]' (7) where a n , b n are undetermined constants. We postpone a discussion of the convergence of the series (eq 7) for t^O and consider it for t=0, where v(x } 0) is supposed to equal f(x): v(x,0) ^^Li(a n cos nx-\-b n smnx). 71 = 1 (8) If the series in eq 8 actually does converge tof(x) for all x, it is shown on p. 274 of [12] that the co- efficients, a n , b n , must be the Fourier coefficients of /. Conversely, by p. 25 of [12], the hypothesis Hi is sufficient to insure that the Fourier series of / actually converges to f(x) ; it even converges uni- formly for x in any interval bounded away from a discontinuity of /. Moreover, the hypothesis H 2 implies that in the Fourier series a =0. We henceforth stipulate that the series (eq 8) is the Fourier series of/. It is important to note 6 that Hi implies that a n and b n are 0(l/n); that is, there exists a constant M <C °° such that \na n \<M, \nb n \<M (all n). 1. Proof of Convergence (9) There remains only a proof that the series in eq 7 actually does converge to a function v(x, t) with the required properties, Pi,P 2 , P 3 , P 4 . It will be useful to have the following representations of cos (tjAn) and sin (tjkn). They are proved by Taylor's formula and hold for all values of t/An: « See p. 18 of [12]. 90 Journal of Research cos-j— =1" An * n t 2 32n 2 where|a„|=a^j|<l; . t U An An where | /S,| =|^(^) ! <1; sin 4n 4n 384n 3 ' (10) (11) (12) where \y n \ =|t(^)| <1. Using eq 10 and 11, one sees that for any fixed t, S «* cos(nx+^J+b n sm(nx+ 4 - ) I S ( a n cos "£+&* sin nx) cos vz + (&„ cos nx— a„ sin ma;) sin ~. = oo S ( a « cos nx+bn sin m) — 71 = 1 t^ (b n a n . \ -7> j p n ( — cos nx sin nx )= 4^ ^ n \n n J (13) Representation as the sum of three series is per- mitted because each of the series 2 , S x , S 2 converges. 2 converges for all x because it is the Fourier series of f ; its convergence is uniform in any interval bounded away from a discontinuity of f(x). Fix any positive number t l9 and restrict the consideration to t'& such that 0<t<ti. Since a n and b n are 0(l/n) 9 2i and 2 2 are convergent uniformly in x and 2. For example, 2 2 is domi- nated by (U/A) ^,(\b n \/n-\-\a n \/ri), a series con- vergent like 2(lM 2 )- The series of eq 7 is thus convergent for all x, t and defines by its limit a function v(x, t): v(x, 0=S &n cos (^+4^)+&n sin ( nx+~ ) • (14) Moreover, the series (eq 14) converges uniformly for x, t such that x is bounded away from a jump of/(x). Since Si and 2 2 converge uniformly, they con- verge to continuous functions of x and t. Thus the only discontinuities of v{x, t) are those from S , that is, those of j(x). This is the property of hyperbolic differential equations that discon- tinuities in their solutions are propagated along characteristics. Let the set of discontinuities of J(x) be denoted by E. For all x, t one may obtain v t by termwise dif- ferentiation of eq 14, because by eq 9 the resulting series is absolutely and uniformly convergent in both x and t: *(*, =*£ g cos (W ±)-<% sin (««+£)} (15) Moreover, v t is continuous for all x, t, so that Pi holds. Now one may not obtain v x by termwise differentiation of eq 14, because the resulting series will generally not converge. However, v x does exist and is a continuous function of x and t for all t and for all x not in E. To see this, one uses eq 10 and 12 to carry the Taylor formula (eq 13) to one higher power of t. It is found that )(Xj t) =f(x) +t ^ ( " 'K n On n )- t 2 32 b n cos nx-\ — 2 sni 7i ^ ■)- 3kS 7 »(«» COS7w: -S 8in7la; } (16) By termwise differentiation of eq 16, it is found that 4- OO #*(#, t) =j r (jJ-tS (a B cos nx+ 6 n sin nx) — 4 71 = 1 t 2 ™ /b n a n . \ . ™ >J a w ( — cos nx sin nx ) + 32 ^i n \n n ) 384 s*& cos tmH — 2 sin nx /'(x)-i#)+2 3 +2, Restricting attention to t with ^ ^ /i< °° , one sees that the expressions 2 3 and S 4 are uniformly convergent with respect to x and t. The series leading to j(x) is uniformly convergent for x in any interval bounded away from a discontinuity Solution of the Telegrapher's Equation 91 of /(a?). Hence, for x not in E, termwise differ- entiation leads to the correct value of v x . More- over, v x is continuous in x and t whenever a; is a point of continuity of f(x). To get v tx one may differentiate eq 15 termwise with respect to x: v tx (x,t) = "*S[ a - co *( nx+ in) +bn sin (^+^)} < 17 ) As remarked after eq 14, the series in eq 17 is uniformly convergent for x, t such that x is bounded away from the discontinuities of f(x). Hence v tx is continuous in x and t for all x, t except for x in E. Since v t , v x , v tx are all continuous, v xt exists and equals v tx except on the lines corre- sponding to the discontinuities of /(a?). This shows that v(x, t) has property P 2 . The eq 17 and 14 show that v satisfies the telegrapher's equation (eq 3). Finally, property P 4 was taken care of by the selection of {a ny b n }. Thus the problem is solved completely. 2. Proof of Uniqueness It will be shown that v(x } t) is the only function that solves the above problem. Suppose that Vi(x,i) were a second solution. Then the differ- ence, w(x,t)=v—Vi, satisfies the same problem with f(x) =0. For each x u by property Pi, w{x u t) and w t (xij t) are continuous functions of t for 0^t<C °°, while w(x, 0) = and w x {x, 0)^=0 are, of course, continuous functions of x. " ' For each t, let the value of w(x, t) be extended as a periodic function of x to all x in the interval [— 2ir, 2ir]. Now the values w{— ir, t) and w(x, 0) are given on two charac- teristics of eq 3. By pp. 21 to 22 of [10] they are therefore sufficient to determine w(x, t) uniquely for all x, t. On the other hand, the values w(t, t) and w(x, 0) are also sufficient to determine w(x,t) for all x, t. Since w(—tt, t) =w (vr, i) and w(x, 0) = w(— x, 0) = 0, it is seen by symmetry that w(x, t) = w(—x,t). Now since the values of x lie on a circle, there is nothing exceptional about the line£=7r. The above argument will also show that, for each value of Xi, w{x\-\-x } t) = w(xi—x, t). It follows that for each fixed t, w(x, t) = constant, whence w(x, t) = hit). Byeq3, —iw=w tx z= (d/dx)h / (t) = 0. Hence, the constant value of w(x, t) must be everywhere zero. Then v=v h and the solution v(x, t) given by eq 14 is unique. The results of section III may be summarized in the following theorem, phrased in the notation of section II. Theorem 1. If the real-valued junction f(x) defined on C satisfies hypotheses Hi and H 2 , then there exists a unique function, v(x, t), defined on R and possessing properties P u P 2 , Pz, and P^ If eq 8 is the Fourier series of f(x), then v(x, t) is defined explicitly by eq 1/+. It is of mathematical interest 7 to note that Theorem 1 can be extended to general functions f(x) of bounded variation. That is, one may re- place Hi by the weaker hypothesis Hi: f(x) is of bounded variation on C. Moreover* f(x)=i[f(x+0)+f(x-0)], (all*). The solution v(x, t) is required to have property P 2 , instead of P 2 : P 2 : v x exists in R except for x in a set E (EcC) of Lebesgue measure zero; for all t and for all x not in E, v tx exists and is a continuous function of x and d t. The extension of Theorem 1 is stated as follows: Theorem 2. If the real-valued function f(x) defined on C satisfies hypotheses Hi and H' 2 , then there exists a unique function v(x, t) defined on R and possessing properties P u P' 2 , P 3 , and P 4 . If eq 8 is the Fourier series of f(x), then v(x, t) is defined explicitly by eq H. The convergence proof of section III, 2 requires only slight modification to serve as a proof of Theorem 2. For an arbitrary function f(x) of bounded variation, there need be no interval of continuity; one may therefore not expect the series (eq 14) to converge uniformly in any interval. The termwise differentiations of section III, 2 can, however, be justified for almost all x by the fact that the resulting series are Fourier series. IV. Representation by a Green's Function The formula (eq 14) for the solution of the problem of section II is directly adapted to numerical computation only when the Fourier coefficients a n) b n converge rapidly to zero. But some of the most important cases in meteorology are where f{x) has discontinuities (see footnote 7). 7 This extension seems to have no meteorological interest. However, it is of much importance in meteorology to deal with functions f{x) with some dis- continuities; such discontinuities occur at fronts between air masses. 8 Same aseq 1. 9 It follows from Pi and P' 2 that v xt exists and equals v tx for all t and for all x not in E; see pp. 55 to 57 of [3]. 92 Journal of Research With such an / the Fourier coefficients are, roughly speaking, of the order 0(1 /n), and for those / the convergence of eq 14 is hopelessly slow. It is possible, however, to improve the con- vergence of eq 14 to such a degree that computa- tion of v(x y t) is reasonably possible. The pro- cedure will be illustrated in section V for one particular choice of f(x) : v sin nx f(x)=<r (x)-- 71-1 (0<x<x) (x=0) ■iU+x) (-w<x<0), (18) It may be shown by direct computation that the Fourier series of oo(x) is the series of eq 18. It then follows that the series converges to a {) (x) for all ./'. The reason for choosing a (x) is two-fold: (a) it is of meteorological interest to see how a simple discontinuity in v(x, 0) is propagated, as t increases; (b) for any f(x) that is sectionally smooth, it is possible to represent the correspond- ing v(x, t) in terms of the solution for the special initial condition v(x, Q)=a (x). The present section is devoted to proving the property (b). Suppose, therefore, that G(x,t) is the solution of the problem of section II with the initial condition <r (x)) then G(x, 0) = a (x). Let a sectionally smooth function f(x) be given that satisfies eq 1 and 2. Let /(#) have the jump J k = f(x k +0)—f(x k —0) at the point x k (k = l, 2, . . . ,K). Let Co Or) be continued periodically for x in [— 2ir, 2w\. Then (J k /ir)a (x—x k ) also has the jump J k at the point x k . Now 1 K £0*0 =f( x ) S Jk<r (x—x k ) 7Tfc=i is a continuous function, since all the jumps have been removed. Moreover, £(x) and the functions (J k /ir)<r (x—x k ) all satisfy eq 1 and 2. There is, therefore, a unique solution to the problem of section II for each of these functions. For the function (J k /Tr)<r (x—x k ) the solution is (Jjc/tt) G(x—x k , t). It will be shown below that the solu- tion y(x, t) corresponding to the initial values £(x) is given by 1 f * y(x,t)=- G(x-u,t)£'(u)du. (19) Since the problem of section II is linear in the initial condition /(#) , and since /(*) =*(*)+- S eWs-z*), (20) it follows that 1 K I r^ v ( Xj t)=-^J k G(x—x k ,t)+- G(x-u,t)Z'(u)du. (21) The formula (eq 21) is the desired representation of v(x, t) in terms of the solution G(x, t) to the single problem where f(x) = o- (a;) . The nature of eq 21 indicates that G(x,t) may be called the Green 7 s function of the problem of section II. The repre- sentation (eq 21) is not only of theoretical impor- tance, but it can also be used for approximating the solutions for general boundary values, fix), once the Green's function is tabulated. The prac- tical problem then becomes one of approximating the integral in eq 21 by some numerical process. This latter problem is not treated here. It remains to prove eq 19. First, it may be observed that, for each x, since <r and £ are periodic, a (x—u)£' (u)du=- a (;x—u)d£(u) = Wj-TT TTj-TT i r *■ i r x ~° , £('u)d u <T (x — u) =- £(u)a (x—u)d TTJ-tt TTJ-tt II i C w + t(x)+- £(u)a (x—u)du= irjx+o 1 f* £ (x) - ^ J _J (u) du=£ (x) . (22) In eq 22 we have used two Riemann-Stieltjes in- tegrals. The last step is true because £(x) satisfies eq 2. By eq 13, G(x, f)=a Q (x) +J S § cos nx-~ J2 °^ sin nx. Hence if* i f 7r t f 71 " r^% "l G(x—u,t)£'(u)du = ~ <r (x—u)£'(u)du-{--7— z_, ~| (cos nx cos nu-{- sin nx sin nu) \£'(u)du— 7T J-7T KJ-Tr 47T J-ttLw=1 n J t 2 f* r " a n/ . . ."!,.,. . . (23) ^T" > i -s (sm nx cos nu— cos nx sin nu) \£(u)du. 6ZT J-7T L W=l n J Solution of the Telegrapher's Equation 93 Since £(u) is sectionally smooth, \£'(u)\ is bounded. Hence the series in eq 23 remain uniformly con- vergent when multiplied by %'(u) and may be inte- grated termwise. We note that - cos nu%'(u)du=- sin nu%(u)du= nb n , 7T J-7T 7T J-7T - $mnu£'(u)du= : cos nu£(u)du= — n a n , (24) where a n and b n are the Fourier coefficients of f. In view of eq 22 and 24, the termwise integra- tion in eq 23 yields - C G(x-u,t)Z'(u)du=ttx) + TTJ-TT t °° B j^Lj— (b n cos nx—a n sin nx) — 4 n= i n OL n ™ 2 d? («n cos w«+ ft« sin nx) . oz w= ] n (25) But, by eq 13 and 14, the right-hand side of eq 25 is y(x,f), the solution corresponding to the initial values £(#). This completes the proof of eq 19. The representations (eq 20 and 21) assume a more symmetric and unified form when the Le- besgue-Stieltjes integral is used. It can be shown that /(*)=-( <T (x-u)df(u), (26) and that »(M)=-f G(x-u,t)df(u), (27) TTJ C where eq 26 and 27 include Lebesgue-Stieltjes in- tegrals over the circle C. Whenever f(x) has a discontinuity (say at x{) , the integral in eq 26 fails to converge as a Kiemann-Stieltjes integral for x=Xi, because the. functions o- (xi—u) and f(u) both have a discontinuity for u=0. The same holds for eq 27. The integrals (eq 26 and 27) are convergent for all x as Lebesgue-Stieltjes integrals. Moreover, the formula (eq 27) yields the solution of the problem when f(x) is an arbitrary function of bounded variation; the above proof of eq 19 can be modified to serve as a proof of eq 27. V. Computation of the Green's Function For the purpose of using the representation (eq 21) and for its own meteorological interest, it was desired to compute the Green's function G(x, t). A tabulation to three decimal places, accurate to approximately 0.001, appeared sufficiently accu- rate. An x interval of 5 degrees of longitude (7r/36 radian) is convenient in meteorology. It was decided to compute G(x, t) for various times t up to 96 hours at latitudes <f> from32°19 / to 10 55°. Since the length unit is here the radian of longitude, the expression d<t>/djjL in section I takes the value cos <j>. Then 40=812 cos 2 <£, where ft=7.292X10" 5 radian/second. Now 24 hours corresponds to r =86,400 seconds, or to t= 50.40 cos 2 <£. At latitude 32°19 / , cos 2 0=0.714, whence £=36 at 24 hours. The largest value of t for which G(x, t) was computed corresponds to 96 hours at latitude 32°19'; it is £=144. Let 2=//4, for convenience. By eq 14 G(x, z) =X1 - sin ( nx-\ — )• (28) In summary, a method is required to compute G(x, z) to an accuracy of approximately ±0.001 for 11 x= — 7r(7r/36)7r and for various positive values of z up to 36. The present section will present one such procedure, an application of a method for improving the convergence of certain Fourier series, given on pp. 84 to 88 of [10]. The 'procedure presented below is not an exact description of the methods actually used in making the table of section VIII. It is assumed in section V that computing machinery is available capable of dealing with numbers of 10 decimal digits, but no more than 10. Of the tolerable error 0.001, the amount 0.0005 must be reserved for round-off in the final tabula- tion to three decimal places. Suppose that 0.0004 is allowed for truncation errors, 12 and 0.0001 for computing errors resulting from round-off s during the calculation with 10-digit numbers. To have a truncation error as low as 0.0004 from use of a partial sum of eq 28 would require about 23,000 !« The limit <£=32°19' arose unintentionally. 11 The notation x=a(8)b means x=a, a-\-8, a+25, a+38, . . . , 6—5, b. 12 Truncation errors are errors that result from use of approximate mathe- matical formulas, e. g., use of partial sums of infinite series. 94 Journal of Research terms for x=7r/36 and #=357r/36. The con- vergence must obviously be improved. 1. Representation by Truncated Double Sum We write 00 1 z G(x, z) = ^Pi — sin nx cos -+ n — l Jl 71 00 I 2 Xj - cos nz sin -=Si+S 2 ; (29) n = l ^ ^ for simplicity we consider only the first sum Si in eq 29. It can be shown that the term S 2 behaves similarly throughout the analysis. Expanding cos (z/ri) in its Maclaurin series, one has v-^ 1 • ^ ( -l) r z 2r _^ ^ (-l) r z 2r sin nx Zl ~ir{ n sm nx fa (2r) !n» -£j fa (2r) to 2 '* 1 (30) One type of truncation of eq 30 consists in omitting all terms for n>N-\-l, r>R. Let the error caused by this truncation be called ei. We shall estimate e x for 0<z<Z: «i = n = A 7 +l r=/t! (— l) r ^ 2r sin ra (2r)!^ 2r+1 '! Jn x « /eZV oo V2r oo 1 oo 72r = s L'r i 1 £4 (2r) (2r)! ^ 4 V* The last step above uses the Stirling expression for the factorial function, which, according t o p. 74 of [8], is a one-sided estimate: s\ > s s e~ s ^2irs. Continuing, one finds for <?Z/2i?iV<l that i i^l f eZV R 1 ^( eZ V r {eil< -±^-\2RNj B*>*fa\2RNj 4V7ri? 3/2 The first estimate in eq 31 seems crude, but it does not affect the values of N or R very much. Thus F(A T ,R) is an upper bound for the trunca- tion error |ei| introduced by omitting terms of type n>N+l, r>R in S lt One can reasonably tolerate a truncation error |ei| of 5X10 -5 . The corresponding admissible values of N and R come from setting F(N,R) = 5X10 -5 for Z=36. The following pairs of values of N,R were obtained from eq 31 by a numerical calculation followed by a round-off of N to an integral value: N=2Q 18 13 11 iJ=4 N=9 7 4 R=S 10 15 20 (32) The selection of the most suitable pair of values N,R from eq 32 will be postponed until we have discussed the summation of the remaining terms of eq 30. 2. Computation of the Double Sum The terms of Si for n=l, 2, . . . , N and all r may be left in the form S 1 =>,( - cos- )si »=I \n n/ sm nx, (33) and may be computed from this formula. The terms for n>N+l and r=0, 1, . . . , R—l may be written in the form where S '~~§(2r)! a * )(x) ' 4fr» = (-iy £^r (34) (35) Once g^ (x) has been tabulated for the one value of N to be selected below, Sg may be computed directly from eq 34. Two methods are needed to get a^ } (x), as the calculating machinery is as- sumed to be limited to ten decimal digits. The first method is to use the identity c%\x) = c 2T {x)-{-iyJt S -^g' (36) where «2r(x)=^(x) = (-iy± S -^- (37) In section VI it will be shown that for 0<x<7r, g 2t {x) is essentially a Bernoulli polynomial in the variable x/2w. Hence <r 2 r(x) can readily be corn- Solution of the Telegrapher's Equation 862453—50- 95 puted or even interpolated from existing tables like [4]. One can then get cr^ix) by carrying out the subtraction indicated in eq 36. The terms a 2r (x) and (sin x)/l of eq 36 are approximately unity; with ten-digit calculating machinery these terms may be carried to ten decimal places. Hence 0-2^ (x)j calculated from eq 36, will be good to ten decimal places. In order to get the Green's function to an accuracy of 0.001, it is necessary that all terms in the sums of eq 33 and 34 be sub- stantially correct to five decimal places; hence the terms z 2r o^ (x) I (2r)\ in eq 34 must be given to five decimal places. It follows that z 2r j(2r)\ must be less than 10 5 ; for 2=36 this means that 2r<4. Thus the first method of computing a^\x) is adequate when 2r=0, 2, 4. For 2r > 4, another way of getting o-^° (x) is needed. For these larger r, the convergence of the infinite series (eq 35) is good, and we may write <^>(^(-ir 2 S 4|?- (38) n=N+l n The truncation error e 2 in eq 38 may be shown to be at its maximum when z=7r/36 and when sin (N 2T x) =0. In this case the error is not greater than the sum of the first 36 omitted terms, which is estimated by ^ 36 s in (rnr/36) . 1 w=AT 2r +l mr ^ 72 ^iV 2 2 ; +1 tA ° ll± 3^7rN 2 2 ; +1 In order to keep € 2 z 2r /(2r)\ numerically less than 5X10 -5 for ^=36, it is sufficient that KfJ +1 (^ =5Xl(r - (39) Solution of eq 39 gives the following points of truncation of eq 38: iV 4 =126, N 5 =78, iV 6 =54, JV 7 =40, ^8=32,^=26,^0=22,.... (40) (The values of N k for odd subscripts k are appro- priate to a parallel analysis of the sum S 2 of eq 29, but not to the present analysis of S x .) If ci N) (x), <rs N) (x), . . . , oiftl.2 (x) are estimated by eq 38, with the values N 2r taken from eq 40, the individual summands of eq 34 will each have a truncation error not exceeding 5X10 -5 . 3. Operational Analysis, Selection of N and R We now estimate the labor involved in comput- ing the Si of eq 29, in order to select that pair of values of N and R from eq 32 that makes the computational work a minimum. The resulting computing procedure will be optimal in a limited sense — i. e., optimal among the one-parameter family of truncations considered in section V, 1. Although the resulting procedure will be perfectly feasible for computation — indeed, it differs only moderately from the procedure actually used to get the tables of section VIII — it cannot be said to be optimal among all procedures for computing the Si of eq 29. For it has been based on a certain type of truncation of a certain double series (eq 30), and on a predetermined assignment of truncation errors to several subcalculations (eq 38). Given only the nature of the computational machinery, to describe an absolutely optimal procedure of getting Si would seem quite beyond the present powers of analysis. It is customary and quite realistic to estimate the cost of a computation by the number of multiplications required. 13 We shall consider the multiplications required to get Si for one value of z and for one value of x. In eq 33 one may ignore the desk computation required to get (l/ri)cos(z/n) and sinnx; there are then essentially N multiplications involved in eq 33. In using eq 36, one may ignore the work of getting <j 2r (x), which is chargeable to basic table development, and count SN multiplications needed in all to get a<$>(z), ff ( ?W, <r { T(x). To get <r™(x) from eq 38, in view of eq 40, requires 54— iV multipli- cations. To get a { %\ <j { \ from eq 40 requires in all approximately (22—4) (30— iV) multi- plications, where 30 is a rough average of the higher values of N 2r in eq 40. To put Sg together by use of eq 34 involves R multiplications. Summarizing, we find that getting Si by the outlined procedure requires for each value of z and each value of x the number of multiplications W 1 (N,R) = -m + 7N+3lR-RN= 151- (7-22) (31 -N). (41) Minimizing Wi(N, R) over the pairs given in eq 32 selects the pair N= 18, 22=5, for which Wi(18, 5) = 125. Assuming that the computation of S 2 in 13 This method is expecially useful with respect to computations on Inter- national-Business-Machines equipment. 96 Journal of Research eq 29 involves the same considerations, we may therefore propose N=IS, R = 5 as being the optimal values to use in getting G(x, z) by eq 33 and 34 for one value of x and one value of z. The number of multiplications will be 2W X (N, R). Getting G(x, z) for 72 values of x and one value of z involves no change of N, R. Since sin nx [cos nx] is an odd [even] function of x, the number of multiplications in getting G(x, z) for all x will be 72W\(N, R). However, getting G(x, z) for several values of z changes the analysis, because the functions o- ( £? (x), once computed, serve for each new z without change. To get Si for 13 values of z and one value of x, for example, will require, in addition to the multiplications in eq 41, only 12iV multiplications from eq 33 and 12R from eq 34. The total number of multiplications will then be W 1Z (N, R) = -66 + l9N+4'3R-NR= 751-(19-i2)(43-2V). The minimum of Wiz(N, R) is 361, and occurs for iV=13, R=Q. The optimal choice of N, R has changed, though not greatly. Since we expect to use 13 values of z, we adopt the values JV=13, R--6. 4. Summary of the Computation Method With the above choice of N and R, the compu- tation of Si may proceed as follows: (a) Compute sin nx. si =§6 cos C) 9i (b) For r=0, 1, 2, compute sin nx n2r+l > where <T2r(x) is computed from section VI. (c) Compute *£ 3) (x)=-f: s (42) ~4 n' ^ 13) (*)=S sin nx i4 n v « = 14 n (43) (d) Compute 5 z 2r ^2 — 2-1 (<) T \ | °2r \ X J' r = () K" 1 ) ' (e) Compute 2i=2){+2j. The number of multiplications involved in get- ting Si for one x-value and for one .:- value is: (a) 13; (b) 39; (c) 69; (d) 6; (e) 0. When getting Si for one x-value and 13 2-values, one adds 156 multiplications to (a) and 72 to (d). The total for 13 2-values is 355 multiplications per x-value. (The slight discrepancy with the number 361 in subsection 3 is due to the rough estimate previously made for step (c).) For all ^-values (essentially 36), one gets a total of 12,780 multiplications to get Si. To get S 2 in eq 29, one follows analogous steps involving o- 2r+ i (x), <r£!%i(x) t etc. There will be approximately 12,750 more multiplications, mak- ing a total of about 25,500 multiplications to get G(x, z) for the 72 x-values and 11 2-values. The total truncation error in getting Sj is bounded by 2X10 -4 . This is divided into four trunca- tion errors of 5X10 -5 , one for each of the three steps in (c), and one for the terms left out of (e). The truncation error for S 2 is also bounded by 2 X 10" 4 , making a total truncation error of 4 X 10~ 4 . The final round-off of the final answer to three decimal places may introduce an error of 5X10 -4 . The third source of error is the accumulation of round-off s from adding five-decimal-place terms. Each term is accurate to 5 X 10" 6 ; with an assumed rectangular distribution these terms have a dis- persion near 3X10" 6 . Each value of G(x, z) is ob- tained from the addition of about 270 such terms. The dispersion a of the sum is therefore about V270X5X10" 6 , or about 8X10" 5 . One may ex- pect the accumulated error to exceed 2.5 a=2X 10~ 4 in only 1.3 percent of the cases. The sum of the three errors is effectively bounded by 11X10" 4 , or slightly more than 0.001. VI. The Polynomials {^(x)} In section V we made use of certain functions a k (x) defined as follows: 0"2 r(*) = (-l) r S sin nx r~ »" (r=0, 1,2,...); «T 2r+I (x) = (-l)'+ 1 S cos nx (r=0,l, 2, ...), (44) Solution of the Telegrapher's Equation 97 The function <r (x) was used in section IV; see eq. 18. For &>0 the series a k (x) in eq. 44 are abso- lutely convergent; hence they represent contin- uous functions. Since a 2r (x) is odd and (r 2r+1 (x) is even, it is necessary to sum the series (eq. 44) only for 0<£<7r. As stated in eq. 18, 7T 1 <3ro(z) = 2 — 2 X ' (0<^<^)- (45) No\ ; cos nx (7, (X) = — 2-1 o~ 71=1 n 2 Jo w=ir^ Hence (Ti(x) = — -w--{-yX — jX 2 > (0<X<7r). Similarly, for 0<^c<x, one finds 1 ,. t \ W I T 2 <T 2 (X) = —^ X + -^X Z - 12 cr 2\ X ) = CM\~~To X "1 90 12 12 x° — 48 or 4 (a;) = 90 36 a X 3 + 48 240 x°- (46) Use has been made of the formulas 2T =1 71 2 = 7r76;S:=i^- 4 = 7r 4 /90. The functions <r k (x) are therefore all poly- nomials. Their use in improving the convergence of Fourier series is pointed out on pp. 84 to 88 of [10]. Although they may be easily tabulated from eq 46, they may also be adapted from exist- ing tables because they are essentially Bernoulli polynomials. Let {B n (x)\ be the Bernoulli poly- nomials given on p. 181 of [4]. Lemma. 14 For 0^x<7r, andk = Q, 1, 2, . . . , (2tt)* +1 (0 **(*)— 2(Jfc+l)! B * +1 Proof: Define the Bernoulli number B n by the relation B n — B n (0) . These are the Bernoulli num- bers used on p. 21 of [8]; they are B =l, Bi=— J, 5 2 =l/6,J5 3 -0,5 4 =-l/30,5 5 =0,5 6 -l/42, Davis uses other notations in [4]. Now fix x in the interval 0<x<ir. For each k=l, 2, 3, . . . , **(*)=* \ <r*-i(&di-" . Jct ^\ 1 sin — 21/ 2 ZAn** 1 ' "After this paper was completed, Professor D. H. Lehmer called the author's attention to the statement of this lemma on p. 65 of N. E. Norlund's Vorlesungen fiber Differenzenrechnung (Julius Springer, Berlin, 1924). Jo '*- i(edf ~2(f+rn The last step is by eq 9 on p. 21 of [8], which is correct except for sign. Hence, letting x=2irt, 2(27r)- k - 1 a k (27rt): s: B, 2(2ir)-* ff ,_ 1 (2«i|)Ai-^^ I (*=l,2 > 3, . . . ). (47) Now define 2(2ir) (, ff_,(27rf) = -B /0\(=-l). Use eq 47 formally to get 2(2r)~ 1 a (2irt): 2(2x)- 1 <to(2^) = -§^-^ 1 (=-<+|)- (48) Note that eq 48 agrees with eq 45 for a Q (x). Hence eq 48 is a correct formula, although it was only derived formally. We now apply formula (eq 47) repeatedly, getting always correct expressions: / n_ 2 /o ,\ B t B\ t B 2 . 2{Ztt) < T iV* t )--w.2\~V.V.~W Of 9 \-3 /o A— ^° ^ Bl ^ ^2 t B 3 . Z(2ir) a 2 (M)- - Q -y ^-yy 2!~" 2! f!~ 3l'' Hence w 1 ^ 2 ^— SflFT^r 2(t + 1) l(2,r)-'-V2.f) = -g ( ^f^7 * fk+V -sCi 1 )^ 1 "- (49) But it follows from the top of p. 188 of [4] that B k+l (t) =S C^r 1 ) Bf-»-i- (50) Comparing eq 49 and 50, we see that B k+1 (t) = -2(k + l)\(2ir)- k - 1 a k (27rt). Let 2irt=x, and the lemma is proved. VII. Solution by a Difference Equation Our first approximate solution of the problem stated in section II consisted of the approximate evaluation of the integral (eq 21) by means of numerical integration formulas, using the approxi- 98 Journal of Research mate values of 0(x, z) tabulated in section VIII below. A second approximate solution of the problem consists in solving with appropriate boundary conditions a difference equation that is closely related to the differential equation (eq 3). The latter method is considered in detail in [6], where proofs may be found; only a summary is given in the present section. For any positive integer 2N, let h=w/2N; let &>0 be arbitrary. A net is formed of all points (Xj t) of form (fih, vk) , where /x and v are integers satisfying the conditions fx+v = (mod 2), \n\<2N, ^>0. (51) Where necessary we extend the net and the func- tional values periodically in x with period 2w. The differential equation (eq 3) is approximated by the difference equation, v(x-\-h, t+k)—v(x—h, t-\-k) = v(x+h, t—k)—v(x—h, t-k)—hkv(x, t). (52) The boundary conditions of the difference-equa- tion problem are prescribed values of v(x, t) on the two rows £=0, t=k. Assume that for t=k, 2>CM)=o, (53) where the sum is extended over all points of the second row of the net. The boundary conditions and eq 52 then determine the value of v(x, t) on the row t=2k up to an additive constant. The additive constant and hence v(x, 2k) are determined uniquely by requiring that eq 53 hold also for t = 2k. Continuing row after row, one thus de- termines v(x, t) over the whole net. Let the function so determined be denoted by v m (x, f); it depends on N, on k, and on the initial values pre- scribed for the first two rows. The problem of [6] is to see whether v iN) (x, t)-^>v(x, t) as iV— > oo . Let the initial values v(x, 0) on the first row be defined by the relation v(x, 0)=f(x), where j(x) is the function of eq 1. Let k be fixed. Then it is possible to choose the initial values v(x y k) on the second row of the net in such a manner that, as Af-^oo } v m (x, t)->v(x,t) for each t of the net and for each x that is an abscissa of continuity oij(x). If k is allowed to vary with N in such a manner that k—>0 as N-^co ) then v {N) (x, t)-^v{x, t) for each t>0 and for each x that is an abscissa of conti- nuity o{J(x) . In neither case may one, in general, expect the convergence to be uniform in x or t. The method referred to for choosing the values v(x, k) on the second row is not an economical one, and in a practical computation one would prefer a cheaper though approximate method. Two things are shown in [6] about the effects of an approximation of the values of v(x, k) : First, they may introduce ultimate instability into the solu- tion. Even though the solution v(x, t) of eq 3 be identically zero, it is possible that for fixed N and x, \imv {N) (x,t) = + oo. Second, the approximation docs not prevent con- vergence of v {N) (x,t) to v(x,t), provided that the error of the approximation of v(x,k) vanishes as iV-^oo. One reasonable way of causing the error to vanish is to let &— >0. These results show that the difference-equation method is a feasible method of solving the problem of this paper. VIII. Table of the Green's Function In this section is tabulated the Green's function G(x, z), as computed in the Computation Unit of the Institute for Numerical Analysis. The value of the time parameter z corresponding to h hours at latitude <j> is 2=0.52502 h cos 2 <j>. (51) (Except for the last digit of the constant, formula (eq 51) can be verified from the introduction to section V.) Meteorological considerations sug- gested that h should be chosen in convenient mul- tiples of 12 hours, and that </> should be 35°, 45°, or 55°. The latitude 32°19' resulted from a numer- ical error by the author. A limited number of pairs of values of h and <j> were selected for the computation; these pairs arc shown in table 1, together with the corresponding values of z determined from eq 51. For each of the 13 values of z given in table 1 (and for z=0) and for x=— x(7r/36)ir, the Green's function G(x, z) is presented in table 2 to 5 deci- mal places. Since 6? (#, z) has a discontinuity of the first kind at x=0, the values c7( — 0, z) } G(0, z), and 67(+0, z) arc all given. In every case, 67(0, 3)=i[G(-0, z) + G(0, z)] and G(+0, z) — G( — 0, z) = 7r. The computational procedure fol- lowed that of section V in general outline, with certain deviations. It was decided to use N=18, i?=6. The auxiliary functions o- ( £? (x) and Solution of the Telegrapher's Equation 99 o"2r+\ 0*0 were computed from formulas like eq 36; formulas like eq 38 were not used. This necessi- tated carrying considerably more than 10 digits, and so the polynomials <r k (x) were first computed to 17 decimal places. Choice of formula (eq 36) was based on the value of getting these tables of Bernoulli polynomials as a byproduct of general interest. To be definite/ 5 let us write G{x, z) = G 1 (x, 2) + G 2 (x, z) , where G 1 (x i z)=T,lsm(nx+^\ (52) and G 2 (x,z) = ^2 -sin(nx-\ — )• Let (b k /dz k )G 2 (x,z) = G ifc) (x,z),tork=0, 1,2, .... Then G {k) (x, 0)=S -tt sin (nx+Ic %)=g k (x) —h k (x), n =iQ n ^ \ zy where 0*0*0 =S ^ffi sin (nx+k | j, By section VI, the functions ^(z) are polynomials. They were generated on an International-Busi- ness-Machines tabulator to 17 decimal places. The values of h k (x) were computed and subtracted from g k (x) to yield G (k) (x, 0). For any z, G 2 (x,z)=Jly ] GM(x,0)+R k (x ) z), where there exists a z x (0<2!<2) such that z h (53) |ff*(*,2)l<- -,K+1 ^ (K+ "(X, g,)- (K+l)! If 0<z<36,2 n /ll!<.33X10 10 . And \G iu \x, Si)|< S«" 12 <1-5X10- 15 . Hence, for 0<z< 36, |fl„(x,g)l< 0.5X10- 5 . 15 This description of the computational procedure was furnished by Gertrude Blanch of the Computation Unit, Institute for Numerical Analysis Once the values of G (k) (x, 0) were obtained it was possible to generate, very easily, the function G(x,z) for any values of z in the range 0<2<36. To summarize, (a) Gi(x, z) was computed from eq (52). (b) The functions G ik) (x, 0) were computed for £==0,1,2, . . . , 10. (c) G 2 (x } z) was obtained from eq (53). (d) Gi(x, z) and G 2 (x, z) were added, to yield G(x, z). The subsidiary computations in (a) and (b) were carried to nine decimal places, those in (c) to at least seven decimal places in the partial products. Table 2 is believed to be accurate to ±0.00002 for all x and all z. The cosine component of G(x, z) was given a final check by use of the fol- lowing formula: 35 oo n - S G{k*lW J z)=^±-sm~= V {z)- fe = -36 v = l v 72/ The check showed a deviation between the sum and p(z), which was never greater than 0.00005. The sine component of G (x, z) was given a partial check by the formula 35 k XI sin -pr- G(Jct/36,z) =2 cos 7-5+ k = -3Q Z 15 2±T- 1 L>+i cos (: 8/18 \ 1_ 4v+l) 4?— 1 cos am Comparable agreement was found. Finally, the table differences with respect to x are very satisfactory. Table 1. Values of the time parameter, z, corresponding to hours, h, at latitude, <f> 4> h 32°19' 35° 45° 55° Hours 12 3. 150 24 9.000 8.455 6.300 4.145 48 18. 000 16. 909 12.600 8.291 72 27. 000 18. 900 96 36. 000 25. 200 100 Journal of Research Table 2. Table of Green's Function G(x, z). Table 2. Table of Green's Function G(x, z). 2 X 0.00000 3.15012 4.14543 6.30024 8.29086 8.45505 9.00000 Degrees -180 -0.00000 0. 32695 1.08321 -0. 16762 -1.34038 -1.26672 -0. 85722 -175 -. 04363 . 39530 1.08444 -.30315 -1.29132 -1.19935 -.74063 -170 -.08727 . 45756 1.07647 -. 43082 -1. 22533 -1.11601 -.61286 -165 -. 13090 . 51370 1.05993 —.54955 -1.14399 -1.01848 -.47630 -1G0 -. 17453 . 56372 1.03552 -.65844 -1.04899 -0. 90869 -.33337 -155 -.21817 . 60768 1.00392 -.75679 -0. 94217 -.78862 -.18646 -150 -.26180 . 64565 0. 96581 -. 84405 -.82542 -. 66032 -.03791 -145 -.30543 . 67775 . 92189 -. 91985 -. 70067 -. 52583 . 11005 -140 -. 34907 . 70413 . 87284 -.98398 -. 56985 -.38717 . 25533 -135 -. 39270 . 72496 . 81933 -1.03636 -. 43487 -. 24630 . 39595 -130 -. 43633 . 74042 . 76202 -1.07704 -. 29757 -. 10512 . 53015 -125 -. 47997 . 75072 . 70154 -1.10620 -. 15975 . 03460 . 65632 -120 -. 52360 . 75611 . 63851 -1.12412 -.02308 .17118 . 77306 -115 -. 56723 . 75679 . 57351 -1.13116 . 11085 . 30309 . 87916 -110 -. 61087 . 75304 . 50712 -1.12779 . 24058 . 42893 . 97363 -105 -. 65450 . 74510 . 43985 -1.11454 . 36479 . 54746 1. 05568 -100 -. 69813 . 73324 . 37221 -1.09199 . 48229 . 65759 1. 12472 -95 -. 74176 .71771 . 30468 -1.06078 . 59203 . 75838 1. 18034 -90 -. 78540 . 69880 . 23769 -1.02 ICO . 69310 . 84906 1. 22233 -85 -. 82903 . 67677 . 17165 -0.97516 . 78475 . 92902 1. 25066 -80 -. 87266 . 65189 . 10695 -.92219 . 86637 . 99779 1.26545 -75 -. 91630 . 62442 . 04392 -.86344 . 93749 1. 05505 1. 26700 -70 -0. 95993 . 59462 -.01712 -.79967 . 99777 1. 10063 1. 25573 -65 -1.O0356 . 56277 -.07590 -.73162 1. 04701 1. 13450 1.23218 -60 -1.04720 . 52910 -.13215 -.66004 1. 08513 1.15674 1.19701 -55 -1.09083 . 49386 -.18566 -.58566 1.11218 1. 16755 1. 15098 -50 -1.13446 . 45730 -.23624 -. 50920 1.12832 1. 16724 1.09494 -45 -1.17810 . 41965 -.28373 -.43136 1. 13378 1.15623 1. 02978 -40 -1.22173 .38113 -.32798 -.35278 1. 12892 1. 13502 0. 95646 -35 -1.26536 . 34197 -.36889 -.27411 1. 11416 1. 10417 . 87600 -30 -1.30900 . 30236 -.40638 -.19596 1. 09002 1.06431 . 78941 -25 -1.35263 . 26250 -.44037 -. 11887 1. 05705 1. 01616 . 69774 -20 -1.39626 . 22259 -.47084 -.04338 1. 01588 0. 96044 . 60203 -15 -1.43990 . 18280 -.49776 .03002 0. 96717 . 89794 . 50333 -10 -1.48353 . 14330 -.52114 . 10089 .91164 . 82944 .40264 -5 -1.52716 . 10426 -.54100 . 16881 . 85000 . 75578 .30096 -0 -1.57080 .06582 -.55736 . 23344 . 78302 . 67778 . 19926 0. 00000 1. 63662 1. 01344 1.80424 2. 35382 2. 24858 1. 77006 +0 1. 57080 3. 20741 2. 58423 3. 37503 3. 92461 3. 81938 3. 34085 5 1. 52716 2. 36366 1. 53355 1.93219 1. 95959 1. 81396 1. 21684 10 1. 48353 1. 62904 0. 66827 0. 87990 0. 63440 0. 47195 -0. 16296 15 1.43990 0. 99385 -.03330 . 14729 -. 20075 -. 36465 -. 98574 20 1. 39626 . 44908 -.59093 -.32689 -. 66889 -. 82505 -1.40257 25 1. 35263 -.01369 -1.02264 -. 59533 -. 87018 -1. 01399 -1. 53397 30 1.30900 -. 40226 -1.34473 -. 70310 -. 88521 -1.01530 -1. 47484 35 1. 26536 -. 72390 -1.57202 -.68848 -. 77804 -0. 89523 -1. 29869 40 1. 22173 -.98534 -1.71788 -. 58382 -. 59873 -. 70520 -1. 06130 45 1. 17810 -1.19282 -1.79437 -.41615 -. 38563 -. 48429 -0. 80387 50 1. 13446 -1.35212 -1.81235 -.20791 -. 16734 -. 26137 -. 55574 55 1.09083 -1.46858 -1.78153 . 02253 . 03555 -.05690 -.33671 60 1.04720 -1.54714 -1.71063 . 26034 . 20903 . 11545 -.15898 65 1. 00356 -1.59233 -1.60740 . 49376 . 34437 . 24746 -.02884 70 0. 95993 -1.60833 -1.47871 .71367 . 43706 . 33521 . 05197 75 . 91630 -1.59898 -1.33066 .91321 . 48585 . 37809 . 08507 80 .87266 -1.56780 -1.16858 1. 08748 . 49200 . 37799 . 07448 z X 0. 00000 3. 15012 4. 14543 6. 30024 8. 29086 8. 45505 9.00000 Degrees 85 . 82903 -1.51800 -0. 99717 1. 23323 . 45862 .33860 .02585 90 . 78540 -1.45251 -. 82049 1.34855 .39011 . 26486 -.05417 95 .74176 -1.37403 -. 64207 1.43272 . 29171 .16247 -. 15846 100 . 69813 -1.28496 -. 46492 1.48592 . 16915 .03753 -.27984 105 . 65450 -1.18751 -.29158 1. 50910 . 02832 -. 10376 -.41132 110 . 61087 -1.08367 -. 12420 1. 50380 -. 12493 -. 25537 -. 54637 115 . 56723 -0. 97523 . 03545 1. 47204 -. 28498 -.41155 -.67907 120 . 52360 -.86377 . 18594 1.41614 -. 44662 -. 56703 -.80424 125 . 47997 -.75074 .32616 1.33870 -.60507 -. 71708 -.91745 130 . 43633 -.63740 . 45522 1. 24244 -.75611 -. 85757 -1.01507 135 . 39270 -.52487 . 57252 1. 13016 -.89609 -.98499 -1.09431 140 . 34907 -.41413 . 67764 1.00467 -1.02197 -1.09649 -1.15312 145 . 30543 -.30603 . 77037 0. 86874 -1.13129 -1.18985 -1.19019 150 . 26180 -.20133 . 85064 . 72505 -1.22217 -1.26346 -1.20491 155 .21817 -.10064 . 91855 . 57617 -1.29329 -1.31627 -1.19725 100 . 17453 -.00450 . 97429 . 42452 -1.34386 -1.34779 -1.16777 165 . 13090 . 08665 1.01819 . 27234 -1.37355 -1.35800 -1.11746 170 .08727 . 17245 1. 05065 .12170 -1.38248 -1.34730 -1.04773 175 . 04363 . 25262 1.07214 -.02551 -1.37114 -1.31650 -0. 96032 180 . 00000 . 32695 1. 08321 -. 16762 -1.34038 -1.26672 -.85722 12.60048 16.91010 18.00000 18.90072 25.20096 27.00000 36.00000 -180 0. 21759 1. 38333 0.88710 -0. 22251 -0. 23708 -0. 52214 0. 75323 -175 . 20848 1.28719 . 68046 -.44845 -. 13949 -.34513 . 66966 -170 .18520 1.17912 .47714 -. 65407 -.04531 -.19161 . 59996 -165 .14768 1.06395 . 28213 -. 83538 . 03742 -.06711 . 55216 -160 .09639 0.94615 . 09946 -.98956 . 10193 .02510 . 53133 -155 . 03224 . 82961 -.0674'; -1.11498 . 14294 .08392 . 53944 -150 -.04345 .71765 -.21599 -1. 21108 . 15684 .11027 . 57545 -145 -.12901 . 61292 -. 34473 -1.27829 .14173 . 10689 . 63557 -140 -. 22253 . 51745 -. 45323 -1.31787 . 09736 . 07795 . 71381 -135 -.32192 . 43256 -.54152 -1.33178 . 02503 . 02877 . 80257 -130 -.42494 . 35899 -.61014 -1.32256 -.07262 -. 03461 . 89326 -125 -. 52932 . 29691 -. 66035 -1.29314 -. 19180 -.10578 . 97703 -120 -. 63276 . 24596 -.69412 -1.24669 -. 32784 -. 17834 1. 04492 -115 -.73306 . 20536 -. 71352 -1.18651 -.47538 -. 24602 1.09064 -110 -.82808 . 17397 -.72062 -1.11592 -. 62883 -.30340 1. 10661 -105 -.91587 . 15038 -.71767 -1.03810 -.78241 -.34564 1. 08871 -100 -. 99463 . 13297 -. 70728 -0. 95602 -. 93051 -. 36894 1. 03429 -95 -1.06278 . 12001 -.69185 -. 87240 -1.06785 -.37053 0. 94265 -90 -1. 11899 . 10975 -. 67337 -. 78959 -1.18968 -.34876 .81504 -85 -1. 16216 . 10043 -. 65362 -. 70959 -1.29191 -. 30309 .65453 -80 -1. 19145 . 09041 -. 63445 -.63398 -1.37124 -.23406 . 46575 -75 -1.20628 . 07817 -. 61745 -. 56396 -1.42518 -. 14319 .25459 -70 -1.20633 . 06238 -. 60358 -. 50029 -1.45213 -.03286 . 02788 -65 -1.19154 . 04192 -. 59343 -. 44339 -1.45134 .09382 -. 20697 -60 -1. 16210 . 01592 -. 58753 -. 39331 -1.42289 .23321 -. 44192 -55 -1. 11840 -.01623 -. 58625 -.34978 -1.36773 .38109 -.67081 -50 -1.06108 -. 05488 -. 58940 -.31226 -1.28738 .53326 -.88534 -45 -0. 99094 -.10011 -. 59636 -. 28000 -1.18407 . 68527 -1.07970 -40 -.90897 - 15176 -. 60651 -.25205 -1.06049 . 83281 -1.24859 -35 -.81632 -. 20939 -. 61926 -.22736 -0.91976 . 97176 -1.38784 Solution of the Telegrapher's Equation 101 Table 2. Table of Green's Function G(x, z). 2 X 12.60048 16.91010 18.00000 18.90072 25.20096 27.00000 36.00000 Degrees -30 -. 71423 -.27239 - 63367 - 20477 -. 76527 1. 09836 -1. 49453 -25 -.60408 -. 33990 -.64843 -.18315 -.60056 1. 20927 -1. 56697 -20 -. 48731 -. 41091 -.66232 -. 16133 -.42926 1. 30167 -1. 60479 -15 -.36540 -. 48428 -.67437 -. 13823 -.25493 1. 37335 -1. 60877 -10 -. 23989 -. 55874 -.68350 -.112S6 -. 08102 1. 42266 -1. 58O80 -5 -.11230 -.63295 -.68839 -.08436 . 08924 1. 44862 -1. 52370 -0 .01585 -. 70554 -.68784 -.05201 . 25293 1. 45084 -1.44108 1. 58665 . 86526 . 88296 1. 51879 1.82373 3. 02164 0. 12972 +0 3. 15744 2. 43605 2. 45375 3. 08958 3. 39452 4. 59243 1. 70O52 5 0. 67142 -0. 81525 -0. 84342 -0. 27249 -0. 37260 0. 54092 -2. 54939 10 -. 51209 -2. 00417 -1.87971 -1. 21362 -.63490 . 27468 -1. 76069 15 -. 85078 -2. 07925 -1. 75697 -0. 97035 . 26636 1.10072 -0. 49519 20 -. 68020 -1.65134 -1. 16076 -. 29000 1. 20809 1.82328 -. 00356 25 -.24015 -1. 09098 -0. 48829 . 4121.3 1. 76793 2. 06387 -. 20922 30 . 30379 -0. 60220 . 05701 . 93791 1.89206 1. 84177 -. 68549 35 . 84267 -.27771 . 39671 1. 22420 1.69102 1. 32871 -1.06815 40 1. 31002 -.13946 . 52770 1.28534 1.32178 0. 71692 -1. 17421 45 1. 67048 -.16770 . 48736 1. 17420 0. 92635 . 15777 -0. 98652 50 1. 91086 -.32146 . 33030 0. 95687 . 60563 -. 25779 -. 59209 55 2. 03312 -. 55228 . 11399 .69716 .41414 -. 49626 -. 11820 60 2. 04891 -.81299 -.11013 . 44820 . 36577 -. 56829 . 31487 65 1. 97550 -1. 06293 -.30155 . 24868 . 44447 -. 51226 . 62341 70 1. 83275 -1.27046 -. 43250 . 12229 . 61561 -. 37956 . 76541 75 1. 64092 -1. 41383 -.48723 . 07901 . 83661 -. 22221 . 74192 80 1.41914 -1. 48076 -.46049 . 11749 1. 06504 -. 08486 . 58453 85 1.18449 -1. 46731 -.35588 . 22780 1. 26435 . 00000 . 34287 90 0. 95137 -1.37647 -. 18386 . 39425 1. 40727 . 01370 . 07206 95 . 73134 -1.21644 . 04076 . 59799 1.47711 -.04915 -. 17737 100 . 53301 -0. 99910 . 30108 .81915 1. 46763 -. 18262 -.36632 105 . 36223 -. 73849 . 57953 1. 03865 1. 38174 -. 37234 -.47053 110 . 22235 -.44954 . 85891 1. 23947 1. 22959 -. 59864 -.48116 115 . 11450 -. 14704 1.12358 1. 40751 1. 02640 -.83951 -. 40310 120 . 03798 . 15526 1.36058 1. 53214 0. 79018 -1.07324 -.25141 125 -.00939 . 44516 1. 55988 1. 60630 . 53965 -1.28065 -. 04972 130 -. 03091 . 71247 1.71412 1. 62645 . 29276 -1. 44613 . 17798 135 -. 03066 . 94918 1.81868 1. 59230 . 06507 -1.55895 . 40643 140 -.01322 1. 14954 1.87196 1 . 50636 -. 13095 -1. 61332 . 61405 145 . 01665 1.31003 1.87498 1. 37347 -.28656 -1. 60827 . 78435 150 . 05419 1.42912 1.83068 1.20024 -.39683 -1. 54718 . 90692 155 . 09488 1. 50714 1.74337 0. 99449 -. 46051 -1. 43694 . 97765 160 . 13453 1. 54593 1.61879 . 76479 -.47971 -1.28710 . 99822 165 . 16945 1. 54859 1 . 46378 . 51991 -.45930 -1. 10889 . 97514 170 . 19650 1. 51918 1 . 28553 . 26848 -.40628 -0. 91426 . 91844 175 .21318 1. 46239 1.09103 . 01857 -.32913 -. 71502 . 84021 180 .21759 1.38333 0. 88710 -.22251 -.23708 -. 52214 . 75323 IX. References [I] J. Charney, A. Eliassen, G. Hunt, Report on a program for numerical weather predicting, paper presented at a meeting of the American Meteorological Society, New York, N. Y., January 28, 1949. [2] J. Charney, manuscript in preparation for Journal of Meteorology on the work reported in [1]. [3] R. Courant, Differential and integral calculus (trans- lated by E. J. McShane) 2 (Blackie and Sons, London and Glasgow, 1936). [4] Harold T. Davis, Tables of the higher mathematical functions, 2 (Principia Press, Bloomington, Ind., 1935). [5] G. E. Forsythe, J. Meteorology 4, 67 (1947). [6] G. E. Forsythe, Manuscript in preparation on the approximate solution by difference equations of the telegrapher's equation with boundary values on only one characteristic. [7] G. E. Forsythe, A solution of the telegrapher's equation with initial conditions on only one characteristic. Preliminary Report. Paper presented at a meeting of American Mathematical Society, Columbus, Ohio, December 30, 1948. [8] Erwin Madelung, Die Mathematischen Hilfsmittel des Physikers, American ed. (Dover Publications, New York, N. Y., 1943). [9] C. G. Rossby and collaborators, J. Marine Res. 2, 38 (1939). [10] J. D. Tamarkin and Willy Feller, Partial differential equations (mimeographed lecture notes, Brown University, Providence, R. I., 1941). [II] A. G. Webster, Partial differential equations of mathematical physics, 2d corrected ed. (Hafner Publishing Co., New York, N. Y., 1947). [12] A. Zygmund, Trigonometrical Series (Monografje Matematyczne, Warsaw-Lwow, 1935). Los Angeles, March 8, 1949. 102 Journal of Research